There's no such thing as a "secure line". Whoever fed you that line has watched too many movies.
Sending faxes is not a good strategy if you need strong security. Anyone who can eavesdrop on any of the phone lines involved, can eavesdrop on the contents of the fax. But a perhaps even bigger risk is that you don't know who is on the other end, how the fax is being handled, or whether your fax is going to the right place. It would be easy to screw up and (without realizing it) inadvertently send your fax to the wrong number. You also don't know what digital copies of the data may be inadvertently retained on your fax machine or their fax machine, or who may have access to that fax machine, or what may happen to it if it gets sold.
If it's "very private" information, I don't know if I'd use fax. I'd have to know more about what the information is or how severe the consequences are for inadvertent disclosure, and who might have an interest in tapping your phones -- but faxes have some inherent security limitations that are difficult to counter.